It’s a bit of both! Certain commands to the car can be done locally via Bluetooth OR via Tesla servers. The tricky bit is that status always comes from the server. If you are on a VPN that is blocked (like I use NordVPN and it is often blocked) then the app can’t get status and as long as it can’t get status it may not even try a local command. It’s unclear to me under what circumstances it does local vs cloud commands, and it may have to do with a Bluetooth LE connection that you can’t really control.
When you don’t have service, or you’re on VPN, it may be worthwhile to try disabling and reenabling Bluetooth. I have had success with this before. If you’re using android, it seems like the widget also uses Bluetooth, so you could try adding the widget to your home screen and using that. You can also try setting the Tesla app to not be power controlled, so it never gets closed.
Either way, there’s a definite engineering problem here that feels like it should be fixed by Tesla. But I can at least confirm that, even in situations with zero connectivity, you should be able to perform basic commands like unlock and open trunk without data service.